Research in Algebraic Combinatorics

该项目的主旨是继续与约翰Shareshian在对称函数理论，枚举组合学和代数几何之间的相互作用的联合工作。 这个项目产生于一类对称函数的研究，出现在各种情况下，如在工作的Carlitz，斯科维尔和沃恩枚举Smirnov的话，在工作的Procesi和斯坦利的代表性对称群的上同调的环面品种与A型根系，和在工作中的调查员和Shareshian对一个q模拟的欧拉多项式。 为了理解这些结构之间关系的基础，研究者和Shareshian提出了一个对这种关系具有深远意义的概括，本项目将对此进行探索。 该理论的推广涉及斯坦利的色对称函数的拟对称细化，Tymoczko的对称群上同调的正则半单Hessenberg品种的A型，和一个q-模拟的推广（由于德马里，Procesi和Shayman）的欧拉多项式。 的关系将提供一个代数几何的方法来攻击斯坦利和Stembridge的长期$e$-positivity猜想色对称函数和一个猜想的调查和Shareshian对单峰的q模拟德马里-Procesi-Shayman广义欧拉多项式。该项目还将继续研究偏序集拓扑学和枚举组合学之间的相互作用。该资助支持的研究是代数组合学，这是一个数学领域，旨在发展组合学（计数，安排和分析具体离散配置的科学）和涉及复杂抽象代数结构的纯数学领域之间的联系。 我们的想法是利用这些联系来获得更深入的见解，并解决组合数学和其他领域的问题。在组合学中研究的离散构型出现在数学、计算机科学、物理学、生物学和工程学的各个领域; DNA序列、系统发育树和通信网络都是离散构型的例子。组合方法在这些领域中发挥着越来越重要的作用。

The main thrust of the project is to continue joint work with John Shareshian on the interplay between symmetric function theory, enumerative combinatorics, and algebraic geometry. This project arose from the study of a class of symmetric functions that appears in various contexts such as in the work of Carlitz, Scoville and Vaughan on enumeration of Smirnov words, in the work of Procesi and Stanley on the representation of the symmetric group on the cohomology of the toric variety associated with the type A root system, and in the work of the investigator and Shareshian on a q-analog of the Eulerian polynomials. In an effort to understand the basis for the relationship between these structures, the investigator and Shareshian have proposed a far-reaching generalization of this relationship, which this project will explore. The conjectured generalization involves a quasisymmetric refinement of Stanley's chromatic symmetric functions, Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A, and a q-analog of a generalization (due to De Mari, Procesi and Shayman) of the Eulerian polynomials. The conjectured relationship would provide an algebro-geometric approach to attacking Stanley and Stembridge's long standing $e$-positivity conjecture for chromatic symmetric functions and a conjecture of the investigator and Shareshian on unimodality of their q-analog of the De Mari-Procesi-Shayman generalized Eulerian polynomials. The project will also continue the work of the investigator on the interplay between poset topology and enumerative combinatorics.The research supported by this grant is in algebraic combinatorics, which is an area of mathematics that seeks to develop connections between combinatorics (the science of counting, arranging and analyzing concrete discrete configurations) and fields of pure mathematics that involve sophisticated abstract algebraic structures. The idea is to use these connections to gain deeper insights and solve problems in combinatorics and in the other fields. The discrete configurations that are studied in combinatorics arise in various fields of mathematics, computer science, physics, biology and engineering; DNA sequences, phylogenetic trees, and communications networks are all examples of discrete configurations. Combinatorial methods are playing an increasingly important role in these fields.